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Find the element at the specified index of a Spirally Filled Matrix
  • Last Updated : 18 Nov, 2020

Given two integers x and y representing the row and column number in a matrix respectively, the task is to find the integer at the cell (x, y) in the Spirally Filled Matrix.

Explanation: 
The size of the matrix change according to the insertion of elements. 
As the row and columns are filled layer – by – layer, a square matrix will be formed as each row and column are added simultaneously in the following manner: 

  1. A square matrix of size 1 x 1 is formed having a single element 1.
  2. A square matrix of size 2 x 2 is formed after spiral insertion of elements {2, 3, 4} in order: 
    {{1, 2}, 
    {4, 3}}
  3. A square matrix of size 3 x 3 is formed after spiral insertion of elements {5, 6, 7, 8} in order: 
    {{1, 2, 9}, 
    {4, 3, 8}, 
    {5, 6, 7}} 
     

Below image denotes a 4*4 matrix generated following the above steps: 
 

Examples: 



Input: X = 2, Y = 3 
Output:
Explanation: 
The element at row 2 and column 3 is 8.

Input: X = 5, Y = 4 
Output: 20 
Explanation: The element at row 5 and column 4 is 20.

Approach: 
To solve the problem, the logic behind the given spirally filled matrix needs to be understood. Following are the possible cases that needs to be considered:

  • Case 1: If y > x & y is odd 
    The element at (x, y) is equal to y2 – x +1

Illustration: 
y = 3, x = 1 
y2 – x +1 = 9 – 1 + 1 = 9 
Element present at (1, 3) = 9.

  • Case 2: If y > x & y is even 
    The element at (x, y) is equal to (y – 1)2 + x

Illustration: 
y = 4, x = 1 
(y – 1)2 + x = 9 + 1 = 10 
Element present at (1, 4) = 10.

  • Case 3: If x ≥ y & x is even 
    The element at (x, y) is equal to x2 – y +1

Illustration: 
y = 1, x = 4 
x2 – y + 1 = 16 – 1 + 1 = 16 
Element present at (4, 1) = 16.

  • Case 4: If If x ≥ y & x is odd 
    The element at (x, y) is equal to (x – 1)2 + y

Illustration: 
y = 2, x = 3 
(x – 1)2 + y = 4 + 2 = 6 
Element present at (3, 2) = 6.

Hence, in order to solve the problem, we need to evaluate and print the result of the equation corresponding to the condition which the given x and y satisfies.

Below is the implementation of the above approach: 

C++




// C++ Program to find the element
// at given position in spirally
// filled matrix
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the
// element at (x, y)
int SpiralElement(int x, int y)
{
    int r;
 
    // If y is greater
    if (x < y) {
 
        // If y is odd
        if (y % 2 == 1) {
            r = y * y;
            return (r - x + 1);
        }
 
        // If y is even
        else {
            r = (y - 1) * (y - 1);
            return (r + x);
        }
    }
 
    else {
 
        // If x is even
        if (x % 2 == 0) {
            r = x * x;
            return (r - y + 1);
        }
 
        // If x is odd
        else {
            r = (x - 1) * (x - 1);
            return (r + y);
        }
    }
}
 
// Driver Code
int main()
{
 
    int x = 2, y = 3;
    cout << SpiralElement(x, y);
    return 0;
}

Java




// Java program to find the element
// at given position in spirally
// filled matrix
import java.util.*;
 
class GFG{
 
// Function to return the
// element at (x, y)
static int SpiralElement(int x, int y)
{
    int r;
 
    // If y is greater
    if (x < y)
    {
 
        // If y is odd
        if (y % 2 == 1)
        {
            r = y * y;
            return (r - x + 1);
        }
 
        // If y is even
        else
        {
            r = (y - 1) * (y - 1);
            return (r + x);
        }
    }
    else
    {
 
        // If x is even
        if (x % 2 == 0)
        {
            r = x * x;
            return (r - y + 1);
        }
 
        // If x is odd
        else
        {
            r = (x - 1) * (x - 1);
            return (r + y);
        }
    }
}
 
// Driver code
public static void main(String[] args)
{
    int x = 2, y = 3;
     
    System.out.println(SpiralElement(x, y));
}
}
 
// This code is contributed by offbeat

Python3




# Python3 program to find the element
# at given position in spirally
# filled matrix
 
# Function to return the
# element at (x, y)
def SpiralElement(x, y):
     
    r = 0
 
    # If y is greater
    if (x < y):
 
        # If y is odd
        if (y % 2 == 1):
            r = y * y
            return (r - x + 1)
         
        # If y is even
        else:
            r = (y - 1) * (y - 1)
            return (r + x)
             
    else:
 
        # If x is even
        if (x % 2 == 0):
            r = x * x
            return (r - y + 1)
         
        # If x is odd
        else:
            r = (x - 1) * (x - 1)
            return (r + y)
     
# Driver code
if __name__ == '__main__':
     
    x = 2
    y = 3
 
    print(SpiralElement(x, y))
 
# This code is contributed by Amit Katiyar

C#




// C# program to find the element
// at given position in spirally
// filled matrix
using System;
  
class GFG{
     
// Function to return the
// element at (x, y)
static int SpiralElement(int x, int y)
{
    int r;
  
    // If y is greater
    if (x < y)
    {
         
        // If y is odd
        if (y % 2 == 1)
        {
            r = y * y;
            return (r - x + 1);
        }
  
        // If y is even
        else
        {
            r = (y - 1) * (y - 1);
            return (r + x);
        }
    }
    else
    {
  
        // If x is even
        if (x % 2 == 0)
        {
            r = x * x;
            return (r - y + 1);
        }
  
        // If x is odd
        else
        {
            r = (x - 1) * (x - 1);
            return (r + y);
        }
    }
}
 
// Driver code
static public void Main()
{
    int x = 2, y = 3;
     
    Console.WriteLine(SpiralElement(x, y));
}
}
 
// This code is contributed by offbeat
Output: 
8








 

Time complexity: O(1) 
Auxiliary Space: O(1)
 

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